Let $f$ be an entire function on $\mathbb{C}$. Show that, if $f(z_n) \to \infty$ for all sequences $z_n$ such that $z_n \to \infty$, then f must be a non-constant polynomial.
Since $\lim_{z \to 0}f(\frac{1}{z})=\lim_{z\to \infty}f(z)=\infty$ Thus $f(1/z)$ has a pole of some order. Thus the Taylor';s series of $f(z)$ had to be finite otherwise it would be an essential singularity. Thus $f$ is a polynomial. IS this correct?